Insurers and federal officials sifting through insurance applications under the health-care law have identified a raft of errors, including missing customers and inaccurate eligibility determinations that mean people may be enrolled in the wrong coverage.

Thousands of insurance applicants from HealthCare.gov—at least one in five at the height of the problems by one estimate—have received inaccurate assignments to Medicaid or to the marketplace for private plans, or have received incorrect denials, people familiar with the...